Improved Stability Estimates and Flight Time Predictions Using Higher-Order Transverse Discontinuity Mapping in Hybrid Dynamical Systems

TL;DR

The paper addresses inaccuracies in first-order transverse discontinuity mapping for hybrid dynamical systems, particularly near grazing where impact timing can be mispredicted. It develops a higher-order TDM that yields a quadratic equation for flight time with a discriminant condition and a positive root $\delta_+$, avoiding divergences and overestimation of post-impact states. A numerical method to obtain higher-order saltation matrices is proposed, enabling accurate construction of monodromy matrices and Floquet multipliers, and the approach is validated on single and pair-impact oscillators through LE and bifurcation analyses. The results enhance stability analyses and trajectory predictions in hard-impact systems and are generalizable to autonomous/non-autonomous and multi-barrier configurations, with implications for reliable design and control of vibro-impact devices.

Abstract

This article emphasizes on inconsistencies in the dynamical estimates obtained by first-order transverse discontinuity mapping (TDM) and direct numerical observations for hybrid dynamical systems. Pitfalls of locally linearizing hybrid nonlinear dynamical systems near discontinuity boundaries are demonstrated along with examples of how such linearization could lead to incorrect estimates of impact occurrences for transverse interactions with a rigid barrier. A higher-order TDM is proposed to overcome this shortcoming, allowing for better analytical estimation of impact occurrence times, state transitions, and, consequently, the evolution of trajectories. The difference in flight times of two closely initiated trajectories in the local neighbourhood of a discontinuity boundary is estimated up to $\mathcal{O}(2)$. The resulting quadratic equation implies that the orbits local to the impacting state, corresponding to a negative discriminant, won't reach the discontinuity boundary. Further, the $\mathcal{O}(2)$ correction terms to the analytical expression of the TDM ensure that the flight time estimates do not diverge for low-velocity impacts near grazing, thereby avoiding overestimation of the mapped state. A numerical method is subsequently developed to estimate a saltation matrix incorporating the proposed higher-order TDM to avoid incorrect impact occurrences. Modifications to the existing algorithms used to numerically quantify local stability, namely the Lyapunov spectra and Floquet multipliers, are proposed. Stability analyses using the proposed higher-order approach are carried out for representative cases of a hard impact oscillator and a pair impact oscillator, with results consistent with numerically obtained bifurcation diagrams.

Figures (19)

• Figure 1: A schematic of phase portraits of two nearby trajectories exhibiting impact at $\Sigma_{2}$. The blue line shows the actual trajectory. The red dashed line denotes the perturbed trajectory. The horizontal black dashed line $\Sigma_1$ denotes the section from which both the trajectories are initiated, and $\Sigma_2$ denotes the discontinuity boundary. The amber line denotes the perturbation vector.
• Figure 2: Periodically forced linear oscillator with barrier placed at $x = \sigma$.
• Figure 3: Figures depicting flight times of perturbations reaching the discontinuity barrier $\Sigma_2$. Perturbations only reach $\Sigma_2$ at the points where the two surfaces meet. The contour $G(\delta, y_1)$ is shown in green, and the blue plane represents $G(\delta, y_1) = 0$. The locus of the intersecting curve $\delta^{\xi}_{\pm}$ is the roots given in \ref{['eq 21']}, and $\delta^{\xi}_{+}$ is the locus of points for which the impacts occur. Figures correspond to impact states $\mathbf{x}_i = [-0.11, -0.0577068]$ and $\mathbf{y}_- = [0.00435243, -0.00115247]$ at $t_i = 3489.83$. The green surface indicates $G(\delta, y_1)$ vs $\delta$ and $y_1$ with (a) norm of $\mathbf{y}_-$ fixed at $||\mathbf{y}_-|| = 0.00450243$ and (c) second component of $\mathbf{y}_-$ fixed at $y_2 = -0.00115247$. Imaginary and real parts of flight time $\delta$ vs $y_1$ with (b) norm of $\mathbf{y}_-$ fixed at $||\mathbf{y}_-|| = 0.00450243$ and (d) second component of $\mathbf{y}_-$ fixed at $y_2 = -0.00115247$. The impact barrier is placed at $\sigma = -0.11$ with system parameters $\xi = 2.0$ and $\omega = 1.8$.
• Figure 4: Imaginary part of $\delta^{\xi}_+$ vs $y_1$ with (a) norm of $\mathbf{y}_-$ fixed at $||\mathbf{y}_-|| = 0.00450243$ and (b) second component of $\mathbf{y}_-$ fixed at $y_2 = -0.00115247$. The cyan region corresponds to the impact zone where perturbations reach the discontinuity boundary provided they satisfy (a) $-0.0045 \leq y_1 \leq 0.00419$ and (b) $||\mathbf{y}_-|| \leq 0.00412$. Results correspond to impact states $\mathbf{x}_i = [-0.11, -0.0577068]$ and $\mathbf{y}_- = [0.00435243, -0.00115247]$ at $t_i = 3489.83$ with system parameters $\xi = 2.0$, $\omega = 1.8$ and $\sigma = -0.11$.
• Figure 5: Phase-portraits demonstrating perturbed orbits not impacting the discontinuity boundary as predicted by the higher-order TDM. Nominal and perturbed trajectories for three different perturbations and barrier distance: $r_0 = 0.0095$ and $\sigma = -0.105$ in (a), $r_0 = 0.007$ and $\sigma = -0.11$ in (c), and, $r_0 = 0.0018$ and $\sigma = -0.1288$ in (e) with their corresponding zoomed-phase portraits in (b), (d) and (f). Trajectories $\mathbf{x}$ and $\mathbf{x} + \mathbf{y}$ are shown in blue and red with initial perturbation $\mathbf{y} = r_0/\sqrt{2}[1,1]$ and parameters set to $\xi = 2.0$, $\omega = 1.8$ and $r = 0.8$.

Theorems & Definitions (4)

• Theorem 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.4